On a posteriori estimation of the approximation error norm for an ensemble of independent solutions

An ensemble of independent numerical solutions enables one to construct a hypersphere around the approximate solution that contains the true solution. The analysis is based on some geometry considerations, such as the triangle inequality and the measure concentration in the spaces of large dimensions. As a result, there appears the feasibility for non-intrusive postprocessing that provides the error estimation on the ensemble of solutions. The numerical tests for two-dimensional compressible Euler equations are provided that demonstrates properties of such postprocessing.